Problem 6 - An Introduction to Psychological Statistics

Question

You hear that college campuses may differ from the general population in terms of political affiliation, and you want to use hypothesis testing to see if this is true and, if so, how big the difference is. You know that the average political affiliation in the nation is \(\mu=4.00\) on a scale of \(1.00\) to \(7.00\), so you gather data from 150 college students across the nation to see if there is a difference. You find that the average score is \(3.76\) with a standard deviation of \(1.52\). Use a 1-sample \(t\) -test to see if there is a difference at the \(\alpha=0.05\) level.

Step-by-Step Solution

Answer

No significant difference; fail to reject the null hypothesis.

Step 1: Define the Hypotheses

First, define the null and alternative hypotheses. The null hypothesis \(H_0\) states that there is no difference in political affiliation between college students and the general population, i.e., \(\mu = 4.00\). The alternative hypothesis \(H_a\) claims there is a difference, i.e., \(\mu eq 4.00\).

Step 2: Determine the Test Statistic

The test statistic for a 1-sample \(t\)-test is calculated using the formula:\[ t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} \]where \(\bar{x} = 3.76\) is the sample mean, \(\mu = 4.00\) is the population mean, \(s = 1.52\) is the sample standard deviation, and \(n = 150\) is the sample size.

Step 3: Calculate the Test Statistic

Using the formula:\[ t = \frac{3.76 - 4.00}{\frac{1.52}{\sqrt{150}}} \]Calculate the value:\[ t = \frac{-0.24}{\frac{1.52}{12.247}} \approx \frac{-0.24}{0.124} \approx -1.935 \]

Step 4: Determine the Degrees of Freedom and Critical Value

The degrees of freedom (df) for a 1-sample \(t\)-test is equal to \( n - 1 \), which is 150 - 1 = 149. For a two-tailed test at \(\alpha = 0.05\), use a \(t\)-distribution table to find the critical value for \(df = 149\). The critical \(t\)-value is approximately ±1.976.

Step 5: Make a Decision

Compare the calculated test statistic \(t \approx -1.935\) with the critical \(t\)-value ±1.976. Since \(-1.935\) is not less than \(-1.976\) or greater than \(1.976\), we fail to reject the null hypothesis.

Step 6: Conclusion

There is not enough statistical evidence to suggest that the average political affiliation of college students differs significantly from the general population mean of 4.00 at the 0.05 significance level.

Key Concepts

1-sample t-test

The 1-sample t-test is a statistical method used to determine whether the mean of a single sample significantly differs from a known population mean. This test is particularly useful when you are dealing with a small sample size, as it accounts for variability with an estimate based on the data at hand. In the college campus example, we are comparing the mean political affiliation score of students to the national average. You calculate the t-value using the formula:

\[ t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} \]

• \( \bar{x} \) is the sample mean.
• \( \mu \) is the known population mean.
• \( s \) is the standard deviation of the sample.
• \( n \) represents the sample size.

This calculation tells you how much the sample mean deviates from the population mean, relative to the variability within the sample.

null hypothesis

The null hypothesis is a hypothesis that proposes there is no significant difference between a specified population parameter and a sample statistic. In hypothesis testing, the null hypothesis is assumed true until evidence suggests otherwise. For the political affiliation exercise, the null hypothesis is stated as:

• \( H_0: \mu = 4.00 \) - implying that the mean political affiliation score of college students is equal to the national average.

The goal of hypothesis testing is to evaluate this hypothesis with statistical evidence. If the data are not sufficient to reject the null hypothesis, we conclude that there is no statistical difference between the sample mean and the population mean.

alternative hypothesis

The alternative hypothesis stands in contrast to the null hypothesis and suggests that there is a statistically significant difference between the sample and the population. It is what researchers aim to prove in an analysis. For our sample of college students, the alternative hypothesis is:

• \( H_a: \mu eq 4.00 \) - indicating that the mean political affiliation score of college students does differ from the average national score.

Encounters with alternative hypotheses are often described as two-tailed if no direction of the difference is specified, as seen in this scenario. This means that any significant deviation from the population mean in either direction is of interest.

statistical evidence

Statistical evidence is the basis upon which we evaluate hypotheses in research. It involves collecting and analyzing data to assess whether the evidence is strong enough to support or reject a hypothesis. In the context of our t-test exercise, a t-value and critical values are calculated and used to determine if there is enough evidence to reject the null hypothesis.

• If the calculated t-value falls beyond the critical t-value (in a two-tailed test), we reject the null hypothesis.
• If it falls within the critical t-values, as it does in this case (since \(-1.935\) is between ±1.976), we do not reject the null hypothesis.

Therefore, the conclusion is that the gathered statistical evidence was not sufficient to indicate a significant difference in political affiliation.